What's a good, fast, and semi-cheap way to train my mage lvl from 76-82+?

Enchanting diamond bolts

Any other suggestions?

You can make a lot of money enchanting air orbs. It's not the fastest XP out there, but it's pretty decent, and good for crafting if you buy the molten glass and blow the orbs yourself.

Head over to the MTA and work on getting the Bones to Peaches spell or some infinity robes. If you are worried about money loss, keep in mind that infinity robes can be sold for a decent price to recoup some of the cost.

 

 

 

Don't do the teleportation, it's the worst exp in the Arena, but enchanting is very fast experience and the alching is basically high alch for points.